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2 3 ArrAys the regular disposition of elements When it comes to understanding just what the common factors are among the many and various aspects of symmetry, the notions of congruence and periodicity take us a long way. Most symmetries present these aspects in one form or another, and the absence of one or the other usually leads to a reduction, or even the lack of symmetry. For instance two like objects, in no particular relation with each other, are merely similar since although they may be congruent they are not arranged in any order 1, opposite. The addition of a third object allows a degree of regularity to come into play, creating the basis of a recognisable pattern 2. So, in its simplest form, symmetry is expressed as a regularly repeating figure along a line below, a series that may easily be extended into an array 3. Obviously, simple arrangements of this kind could in theory be indefinitely extended, but symmetry will be maintained just so long as both the repeating element and the spacing remain consistent. We can recognise array symmetries in many natural formations, from the familiar rows of kernels in sweet corn 4, to the patterns of scales in fish and reptiles 5. And of course such regular arrangements feature in a great deal of human art and artefactsas in the decorated shamans cloak opposite 6. Naturally, there are often functional as well as aesthetic criteria operating in the formation of arrays, which is evident in the sort of patterns created by brickwork and rooftiles 7,8. 1. Mere similarity. 2. A pattern emerges with three elements 3. Symmetrical arrays involve regular spacing. In essence, all symmetries are based on invariance or selfcoincidence. In geometric symmetry, the imagined movement that is necessary to achieve this, whether it involves simple repetition, reflection, or rotation see over, is known as an isometry see Appendix. 4. 5. 6. 7. 8.
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